Algorithms and bioinformatics Comparative genomics Anthony Labarre September 12, 2016

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(1)Algorithms and bioinformatics Comparative genomics. Anthony Labarre. September 12, 2016.

(2) Introduction Permutations. Context and motivations Comparing genomes. Context and motivations. I. Deoxyribonucleic acid: double helix of nucleotides (A, C, G, T);. I. Complementarity (A-T, C-G): one strip is enough;. I. Gene = sequence of nucleotides (that codes for a specific protein);. I. Chromosome = ordered set of genes;. I. Genome = set of chromosomes;. I. Goal: compare genomes;. 2.

(3) Introduction Permutations. Context and motivations Comparing genomes. Context and motivations. I. Biologists are interested in comparing species, for example: I I. in order to classify them; in order to explain evolution by reconstructing scenarios;. I. (Dis)similarity measures are needed;. I. Usually based on the sequenced genomes;.

(4) Introduction Permutations. Context and motivations Comparing genomes. At the nucleotide level I. Most comparisons take place at the nucleotide level;. Example (sequence alignment) S1 : · · · T | S2 : · · · T. C | C. C G. G | G. C. C | A C. A − − C | T G G C. I. Matches, substitutions, insertions and deletions;. I. Correspond to mutations;. A ··· | − A ···. T. 4.

(5) Introduction Permutations. Context and motivations Comparing genomes. At the “gene” level I. Some mutations act on segments of nucleotides;. I. Those large-scale mutations are called genome rearrangements;. I. Sequence alignment becomes unfit;.

(6) Introduction Permutations. Context and motivations Comparing genomes. At the “gene” level I. Some mutations act on segments of nucleotides;. I. Those large-scale mutations are called genome rearrangements;. I. Sequence alignment becomes unfit;. Example (genomes as sequences of segments) (A). (B).

(7) Introduction Permutations. Context and motivations Comparing genomes. At the “gene” level I. Some mutations act on segments of nucleotides;. I. Those large-scale mutations are called genome rearrangements;. I. Sequence alignment becomes unfit;. Example (genomes as sequences of segments) (A). genome rearrangements. (B).

(8) Introduction Permutations. Context and motivations Comparing genomes. Genome rearrangements I. Our problem:. Problem (pairwise genome rearrangement) Input: genomes G1 , G2 , a set S of mutations; Goal: find a shortest sequence of elements of S that transforms G1 into G2 . I. Related, simpler problem: compute the evolutionary distance dS (G1 , G2 ) (i.e. just the length of a shortest sequence);. I. Many variants, depending on how genomes are modelled, what (and how) mutations are taken into account, etc.;. 8.

(9) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Modelling genomes as permutations I. Genomes are seen as permutations if: 1. they form ordered sequences of genes (or other segments), and 2. they only differ by order (no duplications or deletions).. 9.

(10) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Modelling genomes as permutations I. Genomes are seen as permutations if: 1. they form ordered sequences of genes (or other segments), and 2. they only differ by order (no duplications or deletions).. Example (genomes → permutations) (A). (B).

(11) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Modelling genomes as permutations I. Genomes are seen as permutations if: 1. they form ordered sequences of genes (or other segments), and 2. they only differ by order (no duplications or deletions).. Example (genomes → permutations) (A). 1. 2. 3. 4. 5. 6. 7. (B).

(12) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Modelling genomes as permutations I. Genomes are seen as permutations if: 1. they form ordered sequences of genes (or other segments), and 2. they only differ by order (no duplications or deletions).. Example (genomes → permutations) 5. 1. 2. 4. 7. 3. 6. (A). 1. 2. 3. 4. 5. 6. 7. (B).

(13) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Modelling genomes as permutations I. Genomes are seen as permutations if: 1. they form ordered sequences of genes (or other segments), and 2. they only differ by order (no duplications or deletions).. Example (genomes → permutations) 5. 1. 2. 4. 7. 3. 6. (A). 7. (B). genome rearrangements. 1. 2. 3. 4. 5. 6.

(14) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Genome rearrangements for permutations I. Segments can be numbered as we wish, so we assume either genome is the identity permutation ι = h1 2 · · · ni and we wish to sort the other genome:. Problem (genome rearrangement (permutations)) Input: a permutation π in Sn , a set S ⊆ Sn of (per)mutations; Goal: find a shortest sorting sequence of elements of S for π. I. Again, we can also focus on merely computing dS (π) – the length of an optimal sorting sequence;. I. S must generate Sn for any pair of permutations to be a finite distance apart; 14.

(15) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Notation and definitions pertaining to permutations I. Permutations can be written in one- or two-row notation: 1 2 3 4 5 6 π= = h4 1 6 2 5 3i. 4 1 6 2 5 3. 15.

(16) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Notation and definitions pertaining to permutations I. Permutations can be written in one- or two-row notation: 1 2 3 4 5 6 π= = h4 1 6 2 5 3i. 4 1 6 2 5 3. I. We deal exclusively with [n] = {1, 2, . . . , n};. 16.

(17) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Notation and definitions pertaining to permutations I. Permutations can be written in one- or two-row notation: 1 2 3 4 5 6 π= = h4 1 6 2 5 3i. 4 1 6 2 5 3. I. We deal exclusively with [n] = {1, 2, . . . , n};. I. All permutations of [n] with composition form the symmetric group Sn ;. 17.

(18) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Notation and definitions pertaining to permutations I. Permutations can be written in one- or two-row notation: 1 2 3 4 5 6 π= = h4 1 6 2 5 3i. 4 1 6 2 5 3. I. We deal exclusively with [n] = {1, 2, . . . , n};. I. All permutations of [n] with composition form the symmetric group Sn ;. I. Composition: the usual ◦, which means that in π ◦ σ, σ is applied first;. 18.

(19) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Sorting permutations by adjacent exchanges I. Simple operation : exchange any two adjacent elements:. Example. I. 4. 1. 6. 2. 5. 3. 1. 4. 6. 2. 5. 3. So we want to sort a permutation by performing as few such exchanges as possible;. 19.

(20) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Sorting permutations by adjacent exchanges I. Simple operation : exchange any two adjacent elements:. Example. I. I. 4. 1. 6. 2. 5. 3. 1. 4. 6. 2. 5. 3. So we want to sort a permutation by performing as few such exchanges as possible; Solution: I. sorting: use Bubble Sort;. 20.

(21) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Sorting permutations by adjacent exchanges I. Simple operation : exchange any two adjacent elements:. Example. I. I. 4. 1. 6. 2. 5. 3. 1. 4. 6. 2. 5. 3. So we want to sort a permutation by performing as few such exchanges as possible; Solution: I I. sorting: use Bubble Sort; distance: the number of swaps used by Bubble Sort; 21.

(22) The model Exchanges Larger-scale transformations The directed breakpoint graph. Introduction Permutations. A closed formula for the adjacent exchange distance I. An inversion in a permutation is a pair of misplaced elements: (πi , πj ) with i < j and πi > πj ;. I. The permutation graph of π has [n] as vertex set and the inversions of π as edges;. Example (the permutation graph of h4 1 6 2 5 3i) 4 1. 6 2. 5 3.

(23) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. A closed formula for the adjacent exchange distance. Theorem The adjacent exchange distance of π is |Inv (π)| = |E (PG (π))|.. Proof sketch. Each adjacent swap “fixes” at most one inversion, which is equivalent to removing an edge from PG , and we can always find such a move at every step if π 6= ι.. 23.

(24) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Sorting permutations by exchanges I. Simple operation : exchange any two elements:. Example. I. 4. 1. 6. 2. 5. 3. 2. 1. 6. 4. 5. 3. So we want to sort a permutation by performing as few such exchanges as possible;.

(25) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Sorting permutations by exchanges I. Here’s a more complete example:. Example (a sorting sequence for h4 1 6 2 5 3i) 4. 1. 6. 2. 5. 3. 25.

(26) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Sorting permutations by exchanges I. Here’s a more complete example:. Example (a sorting sequence for h4 1 6 2 5 3i) 4. 1. 6. 2. 5. 3. 1. 4. 6. 2. 5. 3. 26.

(27) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Sorting permutations by exchanges I. Here’s a more complete example:. Example (a sorting sequence for h4 1 6 2 5 3i) 4. 1. 6. 2. 5. 3. 1. 4. 6. 2. 5. 3. 1. 2. 6. 4. 5. 3. 27.

(28) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Sorting permutations by exchanges I. Here’s a more complete example:. Example (a sorting sequence for h4 1 6 2 5 3i) 4. 1. 6. 2. 5. 3. 1. 4. 6. 2. 5. 3. 1. 2. 6. 4. 5. 3. 1. 2. 3. 4. 5. 6. 28.

(29) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Sorting permutations by exchanges I. Here’s a more complete example:. Example (a sorting sequence for h4 1 6 2 5 3i). I. 4. 1. 6. 2. 5. 3. 1. 4. 6. 2. 5. 3. 1. 2. 6. 4. 5. 3. 1. 2. 3. 4. 5. 6. It works... but can we do better? 29.

(30) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Sorting permutations by exchanges. I. Our goal: each element should be “at the right place”;. I. Some elements are already where they should be, so they won’t move; Strategy: read permutation from left to right, and:. I. I I. if πi = i, pass; otherwise, exchange πi with i;. 30.

(31) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Sorting permutations by exchanges. I. The algorithm obviously terminates;. I. At every step, we “fix” one or two positions;. I. We use the minimum number of exchanges;. I. On the other hand, we’d like to be able to compute the distance without sorting;.

(32) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Cycles I. Computing the distance requires using the cycles of the permutation;. I. Those cycles are obtained by iterating the permutation’s action on {1, 2, . . . , n}, stopping when all elements have been visited;. Example (cycles of h4 1 6 2 5 3i) 1. 2. 3. 4. 5. 6. 4. 1. 6. 2. 5. 3.

(33) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Cycles I. Computing the distance requires using the cycles of the permutation;. I. Those cycles are obtained by iterating the permutation’s action on {1, 2, . . . , n}, stopping when all elements have been visited;. Example (cycles of h4 1 6 2 5 3i) 1. 4. 2. 3. 5. 6. 6. 5. 3.

(34) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Cycles I. Computing the distance requires using the cycles of the permutation;. I. Those cycles are obtained by iterating the permutation’s action on {1, 2, . . . , n}, stopping when all elements have been visited;. Example (cycles of h4 1 6 2 5 3i) 1. 4. 2. 3. 5. 6. 5.

(35) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Cycles I. Computing the distance requires using the cycles of the permutation;. I. Those cycles are obtained by iterating the permutation’s action on {1, 2, . . . , n}, stopping when all elements have been visited;. Example (cycles of h4 1 6 2 5 3i) 1. 4. 2. 3. 6. 5.

(36) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Disjoint cycle decomposition of permutations I. Each permutation decomposes into disjoint cycles: 1 2 3 4 5 6 π= = (1, 4, 2)(3, 6)(5). 4 1 6 2 5 3. 36.

(37) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Disjoint cycle decomposition of permutations I. Each permutation decomposes into disjoint cycles: 1 2 3 4 5 6 π= = (1, 4, 2)(3, 6)(5). 4 1 6 2 5 3. I. The graph of the permutation π, denoted by Γ(π), pictures this decomposition:. 4. 1. 6. 2. 5. 3. 37.

(38) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Disjoint cycle decomposition of permutations I. Each permutation decomposes into disjoint cycles: 1 2 3 4 5 6 π= = (1, 4, 2)(3, 6)(5). 4 1 6 2 5 3. I. The graph of the permutation π, denoted by Γ(π), pictures this decomposition:. 4 I. 1. 6. 2. 5. 3. The number of cycles of π is written c(π);. 38.

(39) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Disjoint cycle decomposition of permutations I. Each permutation decomposes into disjoint cycles: 1 2 3 4 5 6 π= = (1, 4, 2)(3, 6)(5). 4 1 6 2 5 3. I. The graph of the permutation π, denoted by Γ(π), pictures this decomposition:. 4. 1. 6. 2. I. The number of cycles of π is written c(π);. I. 1-cycles are sometimes omitted;. 5. 3. 39.

(40) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Cycles and sorting I. Cycles of length 1 correspond to sorted elements; all other cycles consist of elements that are misplaced;. 4. 1. 6. 2. 5. 3. 1. 2. 3. 4. 5. 6. I. Sorting comes down to splitting cycles until we only have cycles of length 1;. I. Our algorithm repeatedly splits k-cycles into a 1-cycle and a (k − 1)-cycle;. 40.

(41) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Computing the “exchange distance” exc(·) I. At each step, we can always create a new cycle if π 6= ι, so: exc(π) ≤ n − c(π). 41.

(42) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Computing the “exchange distance” exc(·) I. At each step, we can always create a new cycle if π 6= ι, so: exc(π) ≤ n − c(π). I. And we can’t do better, so: exc(π) ≥ n − c(π). 42.

(43) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Computing the “exchange distance” exc(·) I. At each step, we can always create a new cycle if π 6= ι, so: exc(π) ≤ n − c(π). I. And we can’t do better, so: exc(π) ≥ n − c(π). I. Therefore:. 43.

(44) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Computing the “exchange distance” exc(·) I. At each step, we can always create a new cycle if π 6= ι, so: exc(π) ≤ n − c(π). I. And we can’t do better, so: exc(π) ≥ n − c(π). I. Therefore:. Theorem ([Cayley, 1849]) The exchange distance of π in Sn is n − c(π).. 44.

(45) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Lessons from sorting by exchanges. I I. Note that ι is the only permutation with n cycles; The formula exc(π) = n − c(π) expresses: I. I. I. the difference between the number of cycles we have and the number of cycles we want; and the fact that at each step, we can obtain exactly one new cycle.. This point of view will be crucial to sorting problems;. 45.

(46) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Block-interchanges and transpositions I I. The mutations we observe in evolution (may) act on intervals; We’ll now look at two generalisations of exchanges: 1. block-interchanges; 2. transpositions;.

(47) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Block-interchanges and transpositions I I. The mutations we observe in evolution (may) act on intervals; We’ll now look at two generalisations of exchanges: 1. block-interchanges; 2. transpositions;. I. Block-interchanges exchange two disjoint intervals in a permutation; 1 2 34 5 6 7 8 9. 1 67 5 234 8 9.

(48) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Block-interchanges and transpositions I I. The mutations we observe in evolution (may) act on intervals; We’ll now look at two generalisations of exchanges: 1. block-interchanges; 2. transpositions;. I. Block-interchanges exchange two disjoint intervals in a permutation; 1 2 34 5 6 7 8 9. I. 1 67 5 234 8 9. Transpositions displace an interval of the permutation; 1 2 34 5 6 7 8 9 10. 1 5 67 8 2 3 4 9 10. 48.

(49) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Computing the associated distances I. Does the disjoint cycle technique “work”?. 49.

(50) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Computing the associated distances I. Does the disjoint cycle technique “work”?. I. Unlikely: the following permutation has n/2 cycles of length two, but bid(π) = td(π) = 1:. n 2. +1. n 2. +2. n 2. +3. ···. n. 1. 2. 3. ···. n 2. 50.

(51) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Computing the associated distances I. Does the disjoint cycle technique “work”?. I. Unlikely: the following permutation has n/2 cycles of length two, but bid(π) = td(π) = 1:. n 2. I. +1. n 2. +2. n 2. +3. ···. n. 1. 2. 3. ···. n 2. We’ll need something else since we cannot bound the effect of an operation in that setting;. 51.

(52) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. The “directed breakpoint graph” I. (the term “breakpoint” will be explained later);. I. Let’s build the directed breakpoint graph of π = h4 1 6 2 5 7 3i:.

(53) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. The “directed breakpoint graph” I. (the term “breakpoint” will be explained later);. I. Let’s build the directed breakpoint graph of π = h4 1 6 2 5 7 3i: 0 3. 1. build the ordered vertex set (π0 = 0, π1 , π2 , . . . , πn );. 4. 7. 1 5. 6 2.

(54) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. The “directed breakpoint graph” I. (the term “breakpoint” will be explained later);. I. Let’s build the directed breakpoint graph of π = h4 1 6 2 5 7 3i: 0 3. 1. build the ordered vertex set (π0 = 0, π1 , π2 , . . . , πn );. 4. 7. 1 5. 6 2. 2. add black arcs for every ordered pair (πi , πi−1 (mod n+1) );.

(55) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. The “directed breakpoint graph” I. (the term “breakpoint” will be explained later);. I. Let’s build the directed breakpoint graph of π = h4 1 6 2 5 7 3i: 0 3. 1. build the ordered vertex set (π0 = 0, π1 , π2 , . . . , πn );. 4. 7. 1 5. 6 2. 2. add black arcs for every ordered pair (πi , πi−1 (mod n+1) ); 3. add grey arcs for every ordered pair (i, i + 1 (mod n + 1));.

(56) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. The “directed breakpoint graph” I. (the term “breakpoint” will be explained later);. I. Let’s build the directed breakpoint graph of π = h4 1 6 2 5 7 3i: 0 3. 1. build the ordered vertex set (π0 = 0, π1 , π2 , . . . , πn );. 4. 7. 1 5. 6. 2. add black arcs for every ordered pair (πi , πi−1 (mod n+1) ); 3. add grey arcs for every ordered pair (i, i + 1 (mod n + 1));. 2 DBG (π) decomposes in a unique way into alternating cycles.

(57) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Formal definition of the directed breakpoint graph. Definition ([Bafna and Pevzner, 1998]) The directed breakpoint graph of hπ1 π2 · · · πn i is defined by: 1. an ordered vertex set V = (π0 = 0, π1 , π2 , . . . , πn ); 2. a bicoloured arc set A = AB ∪ AG , where: 2.1 AB = {(πi , πi−1 (mod n+1) ) : 0 ≤ i ≤ n}; 2.2 AG = {(i, i + 1 (mod n + 1)) : 0 ≤ i ≤ n};.

(58) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Circular vs. linear layout I. One can also represent the directed breakpoint graph using a linear layout without affecting the cycle structure: π7 3 π6 7 π55. π0 0. π1 4 1 π2 6π 3. 0 4 1 6 2 5 7 3 8 π0 π1 π2 π3 π4 π5 π6 π7 π8. 2 π4 I Circular → linear: split 0 into 0 and n + 1; I. Linear → circular: merge 0 and n + 1 into 0;. I. (in both cases: adapt arcs accordingly); 58.

(59) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Intuitions behind DBG (π) – 1 I. Both “monochromatic” cycles represent an ordering: 1. the black one represents the one we have (“reality”); 2. the grey one represents the one we want to obtain (“desire”);. 0. 0 3. 4. 7. 3 1. 5. 6 2. 4. 7. 1 5. 6 2. 59.

(60) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Intuitions behind DBG (π) – 2 I. The alternating cycles are a blend of “reality” and “desire”, and we must act on those cycles to turn “reality” into “desire”;. I. When we are done, we have the largest number of cycles; π7 3. π0 0. π6 7 π55. π1 4 1 π2. 2 π4. 6π 3. ι7. ι0 0 7. ι1 1. ι6 6 ι5 5. 2 ι2. 4 ι4. 3ι 3. 60.

(61) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Proving lower bounds. I. One way of proving lower bounds on distances: be optimistic: 1. find out the “best case”; 2. pretend we’re always in that case;. I. Techniques based on breakpoint graphs follow the same spirit as those based on the disjoint cycle decomposition, but less freedom is allowed (details later);. I. Usually: case analysis to determine by how much a parameter of the graph can change with one operation;.

(62) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Lower bounding the block-interchange distance A block-interchange β(i, j, k, `) increases the number of cycles in DBG by at most 2:. πi−1. πi. πj−1. πj. πk−1. πk. π`−1. π`. πi−1. πk. π`−1. πj. πk−1. πi. πj−1. π`. Theorem ([Christie, 1996]) For all π in Sn : bid(π) ≥. n+1−c(DBG (π)) . 2. 62.

(63) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Lower bounding the transposition distance A transposition τ (i, j, k) increases the number of odd cycles in DBG by at most 2: πi−1. πi. πj−1. πj. πk−1. πk. πi−1. πi. πj−1. πj. πk−1. πk. πi−1. πi. πj−1. πj. πk−1. πk. πi−1. πi. πj−1. πj. πk−1. πk.

(64) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Lower bounding the transposition distance A transposition τ (i, j, k) increases the number of odd cycles in DBG by at most 2: πi−1. πi. πj−1. πj. πk−1. πk. πi−1. πj. πk−1. πi. πj−1. πk. πi−1. πi. πj−1. πj. πk−1. πk. πi−1. πj. πk−1. πi. πj−1. πk. πi−1. πi. πj−1. πj. πk−1. πk. πi−1. πj. πk−1. πi. πj−1. πk. πi−1. πi. πj−1. πj. πk−1. πk. πi−1. πj. πk−1. πi. πj−1. πk.

(65) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Lower bounding the transposition distance A transposition τ (i, j, k) increases the number of odd cycles in DBG by at most 2: πi−1. πi. πj−1. πj. πk−1. πk. πi−1. πj. πk−1. πi. πj−1. πk. πi−1. πi. πj−1. πj. πk−1. πk. πi−1. πj. πk−1. πi. πj−1. πk. πi−1. πi. πj−1. πj. πk−1. πk. πi−1. πj. πk−1. πi. πj−1. πk. πi−1. πi. πj−1. πj. πk−1. πk. πi−1. πj. πk−1. πi. πj−1. πk. Theorem ([Bafna and Pevzner, 1998]) For all π in Sn : td(π) ≥. n+1−codd (DBG (π)) . 2.

(66) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. A more general lower bounding technique [Labarre, 2013] I. Recall the “monochromatic” decomposition of DBG (π): 0 3. 0 4. 7. 3 1. 5. 6 2. 4. 7. 1 5. 6 2. 66.

(67) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. A more general lower bounding technique [Labarre, 2013] I. Recall the “monochromatic” decomposition of DBG (π): 0 3. 0 4. 7. 3 1. 5. 6. 4. 7. 1 5. 6. 2. 2. π̂ = (0, 1, 2, 3, 4, 5, 6, 7). π̇ = (0, 3, 7, 5, 2, 6, 1, 4). 67.

(68) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. A more general lower bounding technique [Labarre, 2013] I. Recall the “monochromatic” decomposition of DBG (π): 0 3. 0 4. 7. 1 5. π=. 3. 4. 7. 6. 1 5. 6. 2. 2. π̂ = (0, 1, 2, 3, 4, 5, 6, 7). π̇ = (0, 3, 7, 5, 2, 6, 1, 4). ◦. 68.

(69) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. A more general lower bounding technique [Labarre, 2013] I. Recall the “monochromatic” decomposition of DBG (π): 0 3. 0 4. 7. 1 5. 6 2. π= =. 3. 4. 7. 1 5. 6 2. π̂ π̇ = = (0, 1, 2, 3, 4, 5, 6, 7) ◦ (0, 3, 7, 5, 2, 6, 1, 4) (0, 4, 1, 5, 3)(2, 7, 6). 69.

(70) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. A more general lower bounding technique [Labarre, 2013] I. We have π = π̂ ◦ π̇, with dcd(π) ' G (π); indeed: 0 3. 4. 7. 1 5. 6 2. π= =. π̂ π̇ = = (0, 1, 2, 3, 4, 5, 6, 7) ◦ (0, 3, 7, 5, 2, 6, 1, 4) (0, 4, 1, 5, 3)(2, 7, 6). 70.

(71) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Rearrangements in the setting of π. I. Bottom line: DBG (π) is a permutation π;. I. We can translate the effect of any rearrangement σ on π:. Lemma ([Labarre, 2013]) For all π, σ ∈ Sn : π ◦ σ = π ◦ σ π .. Proof sketch. Use the definition of π and simplify to the wanted expression..

(72) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Conjugating permutations. Definition Let π, σ ∈ Sn ; the conjugate of π by σ is the permutation π σ = σπσ −1 . An easy (well, less tedious) way of computing π σ : 1 2 ··· n σ1 σ2 σ π= −→ π = π1 π2 · · · πn σπ1 σπ2. · · · σn · · · σπn. . 72.

(73) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. A lower bounding theorem I. This in turns yields:. Theorem ([Labarre, 2013]) Let: 1. S ⊂ Sn , with S = {s1 , s2 , . . .}, 2. S 0 = {s1 , s2 , . . .}, 3. C the set of conjugacy classes that intersect S 0 . Then for all π in Sn , every factorisation of π into t elements of S yields a factorisation of π into t elements of C. I. And as an immediate corollary: dS (π) ≥ dC (π) 73.

(74) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. An algorithm for obtaining lower bounds. I. To derive a lower bound on dS (π), we: 1. compute S 0 = {s1 , s2 , . . .}; 2. identify the set C of conjugacy classes that cover S 0 ; 3. use any lower bound on dC (π) as a lower bound on dS (π);. 74.

(75) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Application: lower bounding the block-interchange distance. I. As an example, consider the block-interchange distance:. 75.

(76) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Application: lower bounding the block-interchange distance. I. As an example, consider the block-interchange distance: 1. if β(i, j, k, `) is a block-interchange, then β(i, j, k, `) is a 3-cycle or two disjoint 2-cycles;. 76.

(77) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Application: lower bounding the block-interchange distance. I. As an example, consider the block-interchange distance: 1. if β(i, j, k, `) is a block-interchange, then β(i, j, k, `) is a 3-cycle or two disjoint 2-cycles; 2. so C is the set of all pairs of 2-cycles;. 77.

(78) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. Application: lower bounding the block-interchange distance. I. As an example, consider the block-interchange distance: 1. if β(i, j, k, `) is a block-interchange, then β(i, j, k, `) is a 3-cycle or two disjoint 2-cycles; 2. so C is the set of all pairs of 2-cycles; 3. since dC (π) = (n + 1 − c(π))/2 (Cayley’s theorem), we recover dS (π) ≥ dC (π) = (n + 1 − c(π))/2 =. n + 1 − c(DBG (π)) 2. 78.

(79) Introduction Permutations. The model Exchanges Larger-scale transformations The directed breakpoint graph. References I. Bafna, V. and Pevzner, P. A. (1998). Sorting by transpositions. SIAM Journal on Discrete Mathematics, 11(2):224–240. Cayley, A. (1849). Note on the theory of permutations. Philosophical Magazine Series 3, 34(232):527–529. Christie, D. A. (1996). Sorting permutations by block-interchanges. Information Processing Letters, 60(4):165–169. Labarre, A. (2013). Lower bounding edit distances between permutations. SIAM Journal on Discrete Mathematics, 27(3):1410–1428.. 79.

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